- Opinion dynamics on social networks: Random graphs can be used to accurately model the evolution of opinions in social networks, since they can incorporate complex phenomena such as selective exposure, confirmation bias, biased media signals, and the presence of bots. Random graph theory is also rich enough to accurately model complex networks exhibiting large levels of inhomogeneity, short typical distances, and community structure. A main question of interest is whether we can explain the main mechanisms in which consensus and polarization can occur, as well as identify strategies that could be used to reduce polarization. Some of the problems I am currently working on involve various models for the evolution of opinions on directed complex networks, under different scaling regimes and a variety of model assumptions.
- Directed preferential attachment graphs: These are a popular family of dynamic random graph models that can be used to explain how large complex networks form. Of particular interest is the ability of these models to explain why many real-world networks exhibit scale-free degrees, which appear as a consequence of the tendency of new vertices to prefer to attach to existing vertices that are already very popular (i.e., have large degrees). My interest lies on directed versions of these models and the structural properties of the graphs they produce, including their local weak limits, the size of their strongly connected components, and the distribution of centrality measures such as PageRank.
- Stability of directed networks: Random graphs are used to model real-world complex networks that are either too large to be analyzed directly or can be constantly changing. They are also very useful for determining whether a real graph is likely to have certain properties, such as short distances between vertices or large connected components. For directed graphs, the connectivity properties are a bit more subtle than in the undirected case, however, it is known that for several popular random graph models there is a threshold that determines whether there will exist a strongly connected component containing a positive fraction of all the vertices in the graph. The project I have in mind consists in analyzing a couple of models to try to determine how stable the size of the strongly connected component is with respect to the addition/removal of arcs. Closely related to this question, is whether typical distances between vertices remain stable under the same type of perturbations.